The question is about Euler's Totient function, denoted as $\phi(n)$, and asks to find the value of $\phi(4^n)$.

Euler's Totient Function:

Euler's Totient function $\phi(n)$ counts the number of integers $m$ such that $1 \leq m \leq n$ and $\gcd(m, n) = 1$. The formula for $\phi(n)$ when $n$ is factored into distinct primes $p_1, p_2, \dots, p_k$ is: $\phi(n) = n \left( 1 - \frac{1}{p_1} \right) \left( 1 - \frac{1}{p_2} \right) \cdots \left( 1 - \frac{1}{p_k} \right)$ In this case, the number $n = 4^n$ is a power of 4, which can be expressed as $4^n = (2^2)^n = 2^{2n}$.

Step-by-Step Solution for $\phi(4^n)$:

• The prime factorization of $4^n$ is $4^n = 2^{2n}$.
• Using the formula for Euler's Totient function for powers of a prime: $\phi(2^{2n}) = 2^{2n} \left( 1 - \frac{1}{2} \right) = 2^{2n} \cdot \frac{1}{2} = 2^{2n - 1}$ Thus, the value of $\phi(4^n) = 2^{2n - 1}$.

Answer:

The correct answer is option B: $2^{2n - 1}$.

Would you like further details or explanations?

Here are 5 related questions to explore:

1. How is Euler's Totient function used in cryptography, particularly in RSA encryption?
2. How would $\phi(n)$ change if $n$ has multiple distinct prime factors?
3. Can you derive $\phi(n)$ for composite numbers, such as $n = 12$ or $n = 30$?
4. What is the relationship between Euler's Totient function and prime numbers?
5. How does the inclusion of additional prime factors impact the calculation of $\phi(n)$?

Tip: The Euler's Totient function is multiplicative, meaning that if two numbers are coprime, $\phi(mn) = \phi(m) \times \phi(n)$.